How Many Rolls to Expect a Triple Six in Dice?

[TranscendArcu]

Homework Statement

The Attempt at a Solution

The Fundamental Rule states that, given the six choices of value on a die, the number of possible outcomes for rolling three dice is: $(6)(6)(6) = 6^3 = 216$. Of these 216 possible outcomes, however, there is only one event that has a sequence of three sixes. Thus, given a single roll of the dice, the probability of not seeing a sequence of three sixes is $\frac{215}{216}$. When the dice are rolled n-number of times, the probability of observing at least one sequence of triple sixes can be calculated as one minus the probability of no sequences of three sixes. We can write,

$1 - ( \frac{215}{216})^n = f(n)$.

Let n be 149, and $f(149) = .4991$. Let n be 150, and $f(150) = .5015$. Thus, I conclude that 150 is the minimum n for a favorable outcome.

Is that right?

 

[mighty2000]

I would respond to this forum post by first acknowledging the attempt at a solution and then providing some additional information and clarification.

Firstly, the Fundamental Rule is a valid way to calculate the total number of possible outcomes for rolling three dice. However, it is important to note that not all outcomes are equally likely to occur. In this case, the probability of getting a specific sequence of three sixes is actually 1/216, not 1/6^3 as stated.

Additionally, the formula used to calculate the probability of observing at least one sequence of triple sixes is correct, but it is important to understand that this is the probability of getting at least one triple six in any of the n rolls, not necessarily in a specific roll. This means that the minimum n for a favorable outcome is not necessarily 150, as there is still a chance of not getting a triple six in the first 149 rolls.

Furthermore, it would be helpful to mention that the probability of getting at least one triple six in n rolls approaches 1 as n increases. In other words, the more times the dice are rolled, the higher the chance of observing at least one sequence of triple sixes.

In conclusion, while the reasoning and formula used in the solution attempt are generally correct, there are some important factors to consider and clarify. As scientists, it is important to be precise and thorough in our explanations and to consider all possible scenarios.